A note on algal population dynamics

Abstract This is a contribution to the special issue honoring the late John R. Blake of the University of Birmingham. All three authors had the pleasure of extensive technical interactions with John Blake during his career in the UK, USA and Australia and benefited both professionally and personally from his friendship. John’s work in developing fundamental mathematical solutions for Stokes’ flows and his application of those mathematical tools to analyses of microorganism locomotion led to special new insights into the world of small-scale swimming. This special issue devoted to John’s memory seems an appropriate occasion to present another fluid mechanical challenge associated with microorganisms, namely the dynamics of algal blooms. Though it is a special reduced-order model that is of limited practical value, John would have particularly enjoyed the analytical solution to the dynamics of algae that was presented by Rutherford Aris (1997, Reflections on Keats’ equation. Chem. Eng. Sci., 52, 2447–2455) in a somewhat eccentric paper. We revisit that solution in this paper and present an extension to Aris’ solution that includes sedimentation of the algae. We think that John would have enjoyed this solution and would, in all likelihood, have been able to expand upon it to include other features such as microorganism buoyancy variations (see, e.g. Kromkamp & Walsby 1990; Belov & Giles, 1997, Dynamical model of buoyant cyanobacteria. Hydrobiologia, 349, 87–97; Brookes & Ganf, 2001, Variations in the buoyancy response of Microcystis aeruginosa to nitrogen, phosphorus and light. J. Plankton Res., 23, 1399–1411), the death of algae (see, e.g. Serizawa et al., 2008a, Computer simulations of seasonal outbreak and diurnal vertical migration of cyanobacteria. Limnology, 9, 185–194; Reynolds, 1984, The Ecology of Freshwater Phytoplankton. Cambridge University Press), the swimming of algae (see, e.g. Pedley, 2016, Spherical squirmers: models for swimming micro-organisms. IMA J. Appl. Math., 81, 488–521) and other relevant hydrodynamic matters.